What You Need To Know

  • Open Sets

    An open set is a set of points that no matter where you are, there is "wiggle room" inside the open set.

  • Topological Spaces

    A topological space \(X\) is a set of points that's covered by a collection of open sets.

  • Continuous Functions

    A continuous function \(f\) between two topological Spaces \(X\) and \(Y\) is a function with no sudden jumps. More technically, \(f:X\to Y\) is continuous if for every open set \(U\) in \(Y\), we have \(f^{-1}(U)\) is open in \(X\).

Homeomorphism

Technical Definition

Suppose you have two topological spaces \(X\) and \(Y\). A homeomorphism is a continuous function \( f:X\to Y\) that is both 1-1 (injective) and onto (surjective) and whose inverse \(f^{-1}:X\to Y\) is also continuous.

Example

\(\mathbb{R}\) and \((0,1)\)

This can be a surprising example because the real number line is unbounded, while the open interval \((0,1)\) is confined to a finite region.

Metric Spaces

Topological Spaces with Rules

A topological space that has a distance function is called a metric space. Learn about how a metric is defined, the importance of the triangle inequality, and see an example.
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Topology Course

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